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arXiv · 2409.05825

Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets

Guy C. David·Matthew Romney·Raanan Schul
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Abstract

A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in $\mathbb{R}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set.

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Guy C. David, Matthew Romney, Raanan Schul. 2024-09-09. Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets. https://arxiv.org/abs/2409.05825

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